Game Theory, Alive

314 Pages·2013·3.06 MB·English

Game Theory, Alive
Anna R. Karlin and Yuval Peres
Draft December 3, 2013
Please send comments and corrections to
[email protected] and [email protected]

i
We are grateful to Alan Hammond, Yun Long, G´abor Pete, and Peter
Ralph for scribing early drafts of this book from lectures by Yuval Peres.
These drafts were edited by Liat Kessler, Asaf Nachmias, Yelena Shvets,
Sara Robinson and David Wilson; Yelena also drew many of the figures. We
also thank Ranjit Samra for the lemon figure, Barry Sinervo for the Lizard
picture, and Davis Sheperd for additional figures.
SouravChatterjee, ElchananMossel, AsafNachmias, andShobhanaStoy-
anov taught from drafts of the book and provided valuable suggestions.
Thanks also to Varsha Dani, Kieran Kishore, Itamar Landau, Eric Lei, Mal-
lory Monasterio, Andrea McCool, Katia Nepom, Colleen Ross, Zhuohong
Shen, Davis Sheperd, Stephanie Somersille, and Sithparran Vanniasegaram
for comments and corrections.
The support of the NSF VIGRE grant to the Department of Statistics
at the University of California, Berkeley, and NSF grants DMS-0244479,
DMS-0104073, and CCF-1016509 is acknowledged.
Contents
1 Introduction page 1
2 Two-person zero-sum games 6
2.1 Examples 6
2.2 Definitions 8
2.3 Saddle points and Nash equilibria 10
2.4 Simplifying and solving zero-sum games 11
2.4.1 The technique of domination 12
2.4.2 Summary of Domination 13
2.4.3 The use of symmetry 14
2.4.4 Series and Parallel Game Combinations 17
2.5 Games on graphs 18
2.5.1 Maximum Matchings 18
2.5.2 Hide-and-seek games 20
2.5.3 Weighted hide-and-seek games 22
2.5.4 The bomber and battleship game 25
2.6 Proof of Von Neumann’s minimax theorem 26
2.7 Linear Programming and the Minimax Theorem 31
2.7.1 Linear Programming Basics 32
2.7.2 Linear Programming Duality 34
2.7.3 Duality, more formally 35
2.7.4 The proof of the duality theorem 36
2.7.5 An interpretation of a primal/dual pair 40
2.8 Zero-Sum Games With Infinite Action Spaces∗ 42
2.9 Solved Exercises 49
2.9.1 Another Betting Game 49
3 General-sum games 51
3.1 Some examples 51
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Contents iii
3.2 Nash equilibria 54
3.3 General-sum games with more than two players 58
3.4 Games with Infinite Strategy Spaces 62
3.5 Potential games 64
3.6 The proof of Nash’s theorem 67
3.7 Fixed-point theorems* 69
3.7.1 Easier fixed-point theorems 69
3.7.2 Sperner’s lemma 72
3.7.3 Brouwer’s fixed-point theorem 75
4 Signaling and asymmetric games 86
4.1 Signaling and asymmetric information 87
4.1.1 Examples of signaling (and not) 88
4.1.2 The collapsing used car market 90
4.2 Some further examples 91
5 Other Equilibrium Concepts 94
5.1 Evolutionary game theory 94
5.1.1 Hawks and Doves 95
5.1.2 Evolutionarily stable strategies 97
5.2 Correlated equilibria 100
5.2.1 Existence of Correlated Equilibria 104
6 Adaptive Decision Making 112
6.1 Binary prediction using expert advice 112
6.2 Multiple Choices with Varying Costs 115
6.3 Using adaptive decision making to play zero-sum games 118
6.4 Swap Regret and Correlated Equilibria 120
6.4.1 Computing Approximate Correlated Equilibria 123
7 Social choice 128
7.1 Voting and Ranking Mechanisms 128
7.1.1 Definitions 129
7.1.2 Instant runoff elections 131
7.1.3 Borda count 132
7.1.4 Dictatorship 133
7.2 Arrow’s impossibility theorem 133
7.3 Strategy-proof Voting 135
8 Auctions and Mechanism Design 140
8.1 Auctions 140
8.2 Single Item Auctions 140
8.3 Independent Private Values 142
iv Contents
8.3.1 Profit in single-item auctions 144
8.4 Definitions 145
8.4.1 Payment Equivalence 145
8.4.2 On Asymmetric Bidding 151
8.4.3 More general distributions 153
8.5 Risk Averse bidders 157
8.5.1 The revelation principle 160
8.6 When is truthfulness dominant? 162
8.7 More profit? 162
8.7.1 Myerson’s Optimal Auction 163
8.7.2 Optimal Mechanism 168
8.7.3 The advantages of just one more bidder... 171
8.8 Common or Interdependent Values 171
8.8.1 Second-price Auctions with Common Value 172
8.8.2 English Auctions 176
8.8.3 An approximately optimal algorithm 177
8.9 Notes 179
8.9.1 War of Attrition 179
8.9.2 Generalized Bulow-Klemperer War of Attrition 180
8.9.3 Proof of Theorem 8.4.11 181
8.9.4 Optimal Mechanism 182
9 Mechanism Design 188
9.1 The general mechanism design problem 188
9.1.1 Myerson’s Optimal Auction 191
9.1.2 Optimal Mechanism 196
9.1.3 The advantages of just one more bidder... 199
9.2 Social Welfare Maximization 199
9.3 Win/Lose Mechanism Design 204
9.4 Profit Maximization in Win/Lose Settings 205
9.4.1 Profit maximization in digital goods auctions 206
9.4.2 Profit Extraction 207
9.4.3 A profit-making digital goods auction 208
9.5 Notes: 209
10 Coalitions and Shapley value 211
10.1 The Shapley value and the glove market 211
10.2 The Shapley value 215
10.3 Two more examples 217
11 Stable matching 220
11.1 Introduction 220
Contents v
11.2 Algorithms for finding stable matchings 221
11.3 Properties of stable matchings 222
11.4 A special preference order case 223
12 Interactive Protocols 226
12.1 Keeping the meteorologist honest 226
12.2 Secret sharing 229
12.2.1 A simple secret sharing method 230
12.2.2 Polynomial method 231
12.3 Private computation 233
12.4 Cake cutting 234
12.5 Zero-knowledge proofs 235
12.6 Remote coin tossing 237
13 Combinatorial games 239
13.1 Impartial games 240
13.1.1 Nim and Bouton’s solution 246
13.1.2 Other impartial games 249
13.1.3 Impartial games and the Sprague-Grundy theorem 257
13.2 Partisan games 263
13.2.1 The game of Hex 266
13.2.2 Topology and Hex: a path of arrows* 267
13.2.3 Hex and Y 269
13.2.4 More general boards* 271
13.2.5 Other partisan games played on graphs 272
13.3 Brouwer’s fixed-point theorem via Hex 277
14 Random-turn and auctioned-turn games 283
14.1 Random-turn games defined 283
14.2 Random-turn selection games 284
14.2.1 Hex 284
14.2.2 Bridg-It 285
14.2.3 Surround 286
14.2.4 Full-board Tic-Tac-Toe 286
14.2.5 Recursive majority 286
14.2.6 Team captains 287
14.3 Optimal strategy for random-turn selection games 288
14.4 Win-or-lose selection games 290
14.4.1 Length of play for random-turn Recursive Majority 291
14.5 Richman games 292
14.6 Additional notes on random-turn Hex 295
14.6.1 Odds of winning on large boards under biased play. 295
vi Contents
14.7 Random-turn Bridg-It 296
Bibliography 300
Index 304
1
Introduction
In this course on game theory, we will be studying a range of mathematical
models of conflict and cooperation between two or more agents. We begin
with an outline of the content of this course.
Webeginwiththeclassictwo-person zero-sum games. Insuchgames,
bothplayersmovesimultaneously,anddependingontheiractions,theyeach
getacertainpayoff. Whatmakesthesegames“zero-sum”isthateachplayer
benefits only at the expense of the other. We will show how to find optimal
strategies for each player in such games. These strategies will typically turn
out to be a randomized choice of the available options.
Forexample,inPenaltyKicks,asoccer/football-inspiredzero-sumgame,
one player, the penalty-taker, chooses to kick the ball either to the left or
to the right of the other player, the goal-keeper. At the same instant as the
kick, the goal-keeper guesses whether to dive left or right.
Fig. 1.1. The game of Penalty Kicks.
The goal-keeper has a chance of saving the goal if he dives in the same
direction as the kick. The penalty-taker, being left-footed, has a greater
likelihood of success if he kicks left. The probabilities that the penalty kick
scores are displayed in the table below:
1
2 Introduction
goal-keeper
L R
-
y
t L 0.8 1
nal ker R 1 0.5
e a
p t
For this set of scoring probabilities, the optimal strategy for the penalty-
taker is to kick left with probability 5/7 and kick right with probability 2/7
— then regardless of what the goal-keeper does, the probability of scoring is
6/7. Similarly, the optimal strategy for the goal-keeper is to dive left with
probability 5/7 and dive right with probability 2/7.
In general-sum games, the topic of Chapter 3, we no longer have op-
timal strategies. Nevertheless, there is still a notion of a “rational choice”
for the players. A Nash equilibrium is a set of strategies, one for each
player, with the property that no player can gain by unilaterally changing
his strategy. It turns out that every general-sum game has at least one Nash
equilibrium. Theproofofthisfactrequiresanimportantgeometrictool, the
Brouwer fixed-point theorem.
One interesting class of general-sum games, important in computer sci-
ence, is that of congestion games. In a congestion game, there are two
drivers, I and II, who must navigate as quickly as possible through a con-
gested network of roads. Driver I must travel from city B to city D, and
driver II, from city A to city C.
A D
(1,2)
(3,5) (2,4)
(3,4)
B C
Fig. 1.2. A congestion game. Shown here are the commute times for the
four roads connecting four cities. For each road, the first number is the
commute time when only one driver uses the road, the second number is
the commute time when two drivers use the road.
The travel time for using a road is less when the road is less congested.
In the ordered pair (t ,t ) attached to each road in the diagram below,
1 2
t represents the travel time when only one driver uses the road, and t
1 2
represents the travel time when the road is shared. For example, if drivers I
and II both use road AB, with I traveling from A to B and II from B to A,
then each must wait 5 units of time. If only one driver uses the road, then
it takes only 3 units of time.